This paper is organized as follows: in section II we calculate the lattice relaxation for two sets of GSFE parameters in Ref. [1]} and Ref. [2]}. In section III, we present our effective continuum theory of the TBG for two microscopic tight binding models in Ref. [3]} and [4]}, with the corresponding parameter values listed in Table REF , REF , and REF . We also plot the energy spectra of the continuum effective theories including the remote bands up \(\sim 200\) meV. Their nearly perfect agreement with the spectra from the tight binding models demonstrates the validity of the constructed continuum theories. In section IV, we investigate the properties of the Bloch states of the narrow bands, including the sublattice polarization, Wilson loops, and the p-h asymmetry. As shown in Fig. REF , the p-h asymmetry is dominated by \(w_3\) , a previously overlooked inter-layer contact coupling. Section V studies the exactly flat band limit when including the lattice relaxation induced pseudo magnetic field. Finally, the section VI is devoted to the summary.
